This seems very suspicious to me. Women arrested as prostitutes--which I take for a reasonable facsimile of a lower class background and life below the poverty line, although it's admittedly not perfect--have very clear first and family names going back to the late Middle Ages. (There are also plenty of prostitutes who took on a street name, like "Spanish Nell" and "Clarice Clatterballock.") We also find parish records noting the "rent" that beggars paid to live in, essentially, ramshackle wooden lean-tos up against the chapel wall--beggars like Margaret Kind and Alice de Goldenlant.

Of course this isn't just a medieval story. The youngest woman exiled to Australia with the First Fleet in 1787 wasn't really a woman at all--Elizabeth Hayward, age 13. Progressive-era reforms in the early 1910s focused significantly on children, including securing free lunches at school, establishing scholarship programs for some working class children to attend better schools, and attempting to reform orphanages to reduce (the fear of) infanticide. And in the interwar system of proto-welfare in London, the unemployment benefit for poor families in 1930 was 2s per child per week; widows received 5s per week for one child and 3s/week for additional children. Although this isn't names, it certainly lends credence to the equal value on children. (Interestingly, although the post-1950s reforms raised pensions and unemployment benefits, they also only recognized formally the first child; either some or all benefits for additional children seem to have been eliminated.)

I'm not familiar with this particular of Willis' books, but I'd guess it has more to do with the specific family situation of a "street urchin" than any kind of widespread pattern. Overall, female as well as male children were a crucial part of London society and the targets of reform efforts long before 1940.

I've read Blackout, but I'd have to go looking to remember what Willis wrote about this detail. I do remember it seeming odd at the time.

In real history, however, the chances of even the youngest daughter of ten or twelve not having a name of her own in 20th century London is deeply unlikely. It is, however, entirely possible that a child's birth was not registered, and so she could possibly have no 'official' name, although that still would not have been common. The chances are pretty high that Willis has based Binnie on an example she found in research - a lot of odd practices came to light and got recorded in the disruption of the Blitz.

It was also moderately common until the mid-20th century to give younger siblings the names of deceased older siblings - I've a few of those in my own family tree. And, although I'm drifting from the period of the question here, I've seen a few cases where siblings in the medieval era had the same name, which boggles me.

In Ireland, the name you are commonly known by can be used as a 'legal' name (even if it's an obvious nickname). There've been a few cases of people discovering in their teens or twenties that what they thought was their "real name", and is on all their paperwork, is not what's on their birth certificate - either they've been using a middle name as a first name, or something completely different. This is often to distinguish them at first from the other examples of Mary, Sean, or whatever in the immediate surroundings, and then gets forgotten.

While I personally enjoyed Blackout, there is fairly widespread consensus that it's anything but "historically accurate". For some examples of the numerous historical gaffes, see this post and its comments and this followup